WAEC Mathematics 2010 Paper

Concept:Use the tangent–chord theorem and the angle in a semicircle property.Explanation:The tangent $MN$ touches the circle at $P$.The angle between the tangent $MN$ and the chord $PS$ is given as $55^\circ$.By the alternate segment theorem, the angle in the alternate segment is equal to this angle.So, $\angle PRS = 55^\circ$.From the diagram, $PR$ is a diameter because $O$ is the centre.Therefore, $\angle PSR = 90^\circ$ (angle in a semicircle).In $\triangle PRS$, the sum of angles is $180^\circ$.$\angle RPS + \angle PRS + \angle PSR = 180^\circ$$\angle RPS + 55^\circ + 90^\circ = 180^\circ$$\angle RPS = 180^\circ - 145^\circ = 35^\circ$.Answer:$\angle RPS = 35^\circ$, so the correct option is C.